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B1

Part of 2004 Putnam

Problems(1)

Putnam 2004 B1

Source:

12/11/2004
Let P(x)=cnxn+cn−1xn−1+⋯+c0P(x)=c_nx^n+c_{n-1}x^{n-1}+\cdots+c_0 be a polynomial with integer coefficients. Suppose that rr is a rational number such that P(r)=0P(r)=0. Show that the nn numbers cnr,cnr2+cn−1r,cnr3+cn−1r2+cn−1r,…,cnrn+cn−1rn−1+⋯+c1rc_nr, c_nr^2+c_{n-1}r, c_nr^3+c_{n-1}r^2+c_{n-1}r, \dots, c_nr^n+c_{n-1}r^{n-1}+\cdots+c_1r are all integers.
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